misfitdislocations-another-method

Characterizing misfit dislocations at interfaces: Yet Another Method!

Kedarnath Kolluri, M. J. Demkowicz

Financial Support:

Center for Materials at Irradiation and Mechanical Extremes (CMIME) at LANL,

an Energy Frontier Research Center (EFRC) funded by

U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences

Acknowledgments: A. Kashinath, A. Vattré, B. Uberuaga, X.-Y. Liu, A. Caro, and A. Misra

Classifying interfaces:Coherent, semi-coherent, and incoherent boundaries

simplified side view

• Lower and upper grains are in “perfect” alignment always

1 4 8 12

1 4 8 13

• Lines of atoms are aligned perfectly only periodically

Classifying interfaces:Coherent, semi-coherent, and incoherent boundaries


Coherent, semi-coherent, and incoherent boundaries

• Atomic interactions generally reduce the “bad” patch

• Coherent region experiences strain emanated by the “bad” patch

• Interface with well separated “bad” patches may be described within

the same theory as that of dislocations: misfit dislocations


Semi-coherent interfaces (2D defects) can be represented asarrays of dislocations (1D defects)

Line defects in metals: Edge dislocation

Defects in Crystals, H. Foell. http://www.tf.uni-kiel.de/matwis/amat/def_en

http://www.tf.uni-kiel.de/matwis/amat/def_en



Line defects in metals: Screw dislocation



Dislocations

screw dislocation edge dislocation

• Dislocation

• has a core (linear elasticity is inapplicable)

• has a line vector (1-d defects)

• described by a vector that displaces atoms when it moves




Semi-coherent interfaces

Semi-coherent interfaces (2D defects) can be represented asarrays of dislocations (1D defects)

General features of semicoherent fcc-bcc interfaces

Cu-V

〈110〉Cu〈111〉Nb

〈112〉 Cu〈112〉 Nb

An example of a semicoherent interface

View of the Interface


Cu-V

〈110〉Cu〈111〉Nb

〈112〉 Cu〈112〉 Nb

An example of a fcc-bcc semicoherent interface

Patterns corresponding to periodic “good” and “bad” regions


Cu-V

〈110〉Cu〈111〉Nb

〈112〉 Cu〈112〉 Nb

Interface contains arrays of misfit dislocations separating coherent regions


〈110〉Cu〈111〉Nb

〈11

2〉C

u〈

112〉

Nb

Cu-Nb Cu-V

Interface contains arrays of misfit dislocations separating coherent regions

Cu-Nb KS Cu-V KS〈110〉Cu

〈11

2〉C

u1 nm

MDI

• Two sets of misfit dislocations with Burgers vectors

• Misfit dislocation intersections (MDI) where different sets of dislocations meet


Sideviews often used to identify dislocation spacing

In this case, dislocation spacing is 10 Cu interatomic plans (2.55 nm)

13

<112>Cu || <112>Nb

1 2 3 4 5 6 7

1 2 3 4 5 6

<110>Cu || <111>Nb

1 2 3 4 5 6 7 8 9 10

1 2 3 4 5 6 7 8 9

d = 2.55 nm CuNb KS

In this case, dislocation spacing is 7 Cu interatomic plans (1.24 nm) 13

<112>Cu || <112>Nb

1 2 3 4 5 6 7

1 2 3 4 5 6

<110>Cu || <111>Nb

1 2 3 4 5 6 7 8 9 10

1 2 3 4 5 6 7 8 9

d = 1.785 nm CuNb KS

12

<112>

Cu |

| <

112>

Nb

<110>Cu || <111>Nb

<111>Cu ||

<110>Nb

2.1 nm

0.9 nm

Side-views by themselves often tell wrong things!

• Misfit dislocations in this case are not perpendicular to the sideview

• The spacing obtained from sideview is not dislocation spacing!

Yet another method for finding dislocations at fcc-bcc interfaces

First, map the atoms in adjacent grains

Ri

R'i

Compute vectors R'i-Ri where the vectors are the lines joining the closest fcc and bcc atoms to their corresponding nearest neighbors

11

1

2

2

3

3

4

4

5

5

6

6

Cu-Fe NW

R0

• Pick an atom in 1 grain. Find the closest 2nd grain atom to that atom

• Take nearest in-grain neighbors around each atom

• Find one-to-one correspondence so that closest atoms are paired!

First, map the atoms in adjacent grains

• R0 -vector between center atoms

• Ri -vector between a center and ith in-grain atom for 1st grain

• R’i -vector between a center and ith in-grain atom for 2nd grain

Ri

R'i


11

1

2

2

3

3

4

4

5

5

6

6

Cu-Fe NW

R0

Find the correspondence matrix that relates R and R’

Ri

R'i


11

1

2

2

3

3

4

4

5

5

6

6

Cu-Fe NW

R0R0

i , Ri

• R and R’ are matrices containing all vectors Ri and R’i

• Solve for R = DR’; D is identify when the locality is coherent

• |D-I| is an intensity indicator - higher the value, lesser the coherency

||R0iR

�1i � 1||

D=I for perfect system like the one here

Adjacent planes of fcc (Cu)

1

2

34

5

6

C

1

2

34

5

6

C

An example result 0.3

0.35

0.4

0.45

0.5

0.55

0.6 0.3 0.35 0.4 0.45 0.5 0.55 0.6

fcc<

112>

fcc<110>

NW

Cu-Fe

0.05

0.1

0.15

0.2

0.25

0.3CuFe NW

|D-I|

Cu<110>

Cu<

112>

• Identifies dislocations well!

• But do information about the characteristics of the dislocation!

Structure of interfaces: Misfit dislocations

• A general method to identify dislocation line and Burgers vectors

• Assumption: A coherent patch exists at the interface

• Advantage: Reference structure not required

• Limitations: Dislocation core thickness cannot be determined (yet)

Identifying the Burgers vectors:

• Take Ri -R’i and make the origin of this vector to the center atom of one grain (any grain)

Ri

R'i


11

1

2

2

3

3

4

4

5

5

6

6

Cu-Fe NW

R0

The computed vectors are plotted. The vectors all originate at the location of the center atom shown in slide 1

fcc<110>

fcc<

112>

First, take Ri-R’i and place it about the center atomCuFe NW

Green: Gradual change in the vectors directionsBlue and Red: Discontinuity in vectors directions(Mean of these vectors are taken as first approximations)

fcc<110>

fcc<

112>

First, take Ri-R’i and place it about the center atom

CuFe NW

Green: Gradual change in the vectors directionsBlue and Red: Discontinuity in vectors directions(Mean of these vectors are taken as first approximations)

fcc<110>

fcc<

112>

First, take Ri-R’i and place it about the center atom

• Take the mean of all the vectors about a single center

CuFe NW

Reduce dimensions by simple average of vectors

• Take average of local deviations (differentiating) of the vectors

CuFe NW

The directions give the Burgers vectors

Reduce dimensions by simple average of vectors

CuFe NW

Atoms are colored by the vector orientation with x-axis

Ang

le o

f th

e B

urg

ers

vecto

r w

ith

X-a

xis

Yellow and light blue: BV is 180 or 0 degrees with +x-axisPurple : BV is 60 degrees with +x-axis Orange : BV is 120 degrees with +x-axis

CuFe NW

Atoms are colored by the vector orientation with x-axis

Ang

le o

f th

e B

urg

ers

vecto

r w

ith

X-a

xis

Yellow and light blue: BV is 180 or 0 degrees with +x-axisPurple : BV is 60 degrees with +x-axis Orange : BV is 120 degrees with +x-axis

• We assumed that the coherent patch is where central atoms overlap

• That assumption may be incorrect.

• We sample other places in the interface and compare with |D-I| plot

CuFe NW

0.3

0.35

0.4

0.45

0.5

0.55

0.6 0.3 0.35 0.4 0.45 0.5 0.55 0.6

fcc<

112>

fcc<110>

NW

Cu-Fe

0

50

100

150

0.3

0.35

0.4

0.45

0.5

0.55

0.6 0.3 0.35 0.4 0.45 0.5 0.55 0.6

fcc<

112>

fcc<110>

NW

Cu-Fe

0

50

100

150

0.3

0.35

0.4

0.45

0.5

0.55

0.6 0.3 0.35 0.4 0.45 0.5 0.55 0.6

fcc<

112>

fcc<110>

NW

Cu-Fe

0

50

100

150

Comparing various possible coherent patchs with |D-I|

0.3

0.35

0.4

0.45

0.5

0.55

0.6 0.3 0.35 0.4 0.45 0.5 0.55 0.6

fcc<

112>

fcc<110>

NW

Cu-Fe

0.05

0.1

0.15

0.2

0.25

0.3CuFe NW

|D-I| map

K. Kolluri, and M. J. Demkowicz, unpublished

Example results and limitations!

• A general method to identify dislocation line and Burgers vectors

• Assumption: A coherent patch exists at the interface

• Advantage: Reference structure not required

• Limitations: Dislocation core thickness cannot be determined (yet)0

0

0.2

0.4

0.6

0.8 1

0 0

.2 0

.4 0

.6 0

.8 1

0 50

10

0

15

0

0

0.2

0.4

0.6

0.8 1

0 0

.2 0

.4 0

.6 0

.8 1

0.1

5

0.2

0.2

5

0.3

0.3

5

0.4

0.4

5

0.5

0

0

0.2

0.4

0.6

0.8 1

0 0

.2 0

.4 0

.6 0

.8 1

0 50

100

150

0

0.2

0.4

0.6

0.8 1

0 0

.2 0

.4 0

.6 0

.8 1

0.0

5

0.1

0.1

5

0.2

0.2

5

0.3

0.3

5

0.4

0.4

5

0.5

0.5

5

Cu-N

b KS

Cu-Fe N

WC

u-V KS

1 nm

0

0

0.2

0.4

0.6

0.8 1

0 0

.2 0

.4 0

.6 0

.8 1

0 50

100

150

0

0.2

0.4

0.6

0.8 1

0 0

.2 0

.4 0

.6 0

.8 1

0.0

6

0.0

8

0.1

0.1

2

0.1

4

0.1

6

0.1

8

0.2

0.2

2

0.2

4

0.2

6

0.2

8

0

0.2

0.4

0.6

0.8 1

0 0

.2 0

.4 0

.6 0

.8 1

0.6

0.8

1 1.2

1.4

0

0.2

0.4

0.6

0.8 1

0 0

.2 0

.4 0

.6 0

.8 1

0.6

0.8

1 1.2

1.4

0

0.2

0.4

0.6

0.8 1

0 0

.2 0

.4 0

.6 0

.8 1

0.6

0.8

1 1.2

1.4

1 nm1.4 nm

Formation energy (eV) Angle with -ve x axis

0 50 100 150Angle with -ve x-axis

misfitdislocations-another-method

Education